Finite Group Databasebeta

← #10897639

Subgroups — #10897639

Order 768.

708 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83811C211yesmaximal, characteristic, complemented
2#78619C211yesmaximal, characteristic, complemented
3C3×#1891C211yesmaximal, characteristic, complemented
4#72028C211yesmaximal, characteristic, complemented
5#71715C211yesmaximal, characteristic
6#71711C211yesmaximal, characteristic, complemented
7#70144C211yesmaximal, characteristic, complemented
8#70130C211yesmaximal, characteristic, complemented
9#69992C211yesmaximal, characteristic, complemented
10#69987C211yesmaximal, characteristic, complemented
11#69965C211yesmaximal, characteristic, complemented
12#69033C211yesmaximal, characteristic, complemented
13#69024C211yesmaximal, characteristic, complemented
14#69017C211yesmaximal, characteristic, complemented
15#69012C211yesmaximal, characteristic, complemented
16#29384—31nomaximal, Sylow
17C3×#568C2211yescharacteristic
18C3×#525C2211yescharacteristic
19#5822C2211yescharacteristic
20#5791C2211yescharacteristic
21C22×C4×Dic3C2211yescharacteristic
22#5745C2211yescharacteristic
23C3×#413C2211yescharacteristic
24C3×#407C2211yescharacteristic
25C3×#407C2211yescharacteristic
26#5236C2211yescharacteristic
27C2×#837C2211yescharacteristic
28C2×#837C2211yescharacteristic
29#5207C2211yescharacteristic
30#5207C2211yescharacteristic
31#5145—22no—
32#5144—22no—
33C2×#825C2211yescharacteristic
34C2×#825C2211yescharacteristic
35#5132C2211yescharacteristic
36#5125C2211yescharacteristic
37#5124C2211yescharacteristic
38#5124C2211yescharacteristic
39#5118C2211yescharacteristic
40#5118C2211yescharacteristic
41Dic3×M16C2211yescharacteristic
42Dic3×M16C2211yescharacteristic
43#5005C2211yescharacteristic
44#4719—22no—
45#4718—22no—
46C3×#344C2211yescharacteristic
47C3×#344C2211yescharacteristic
48C3⋊#338C2211yescharacteristic
49#4559C2211yescharacteristic
50#4556C2211yescharacteristic
51#4555C2211yescharacteristic
52#4555C2211yescharacteristic
53#4532C2211yescharacteristic
54#4532C2211yescharacteristic
55#4531C2211yescharacteristic
56#2956—31no—
57#2920—31no—
58#2102—31no—
59#2091—31no—
60#2050—31no—
61#2010—31no—
62#1932—31no—
63#1932—31no—
64#1896—31no—
65#1896—31no—
66#1891—31no—
67#1891D311yescharacteristic, complemented
68#1861—31no—
69#1861—31no—
70#1841—31no—
71C2×C3×#41C2311yescharacteristic
72C23×C3×C4C2311yescharacteristic
73C23×Dic3—21no—
74C2×#235C2311yescharacteristic
75C3×#131—22no—
76C3×#131—41no—
77C2×C3×M16—41no—
78C2×C3×M16C2311yescharacteristic
79C2×C3×M16C2311yescharacteristic
80C2×C3×M16C2311yescharacteristic
81C2×C3×M16C2311yescharacteristic
82C3×#123—21no—
83C3×C22⋊Q8—21no—
84C3×#121—21no—
85C3×#120—21no—
86C3×C4×D4—41no—
87C3×#117C2311yescharacteristic
88#953—21no—
89#951—21no—
90#950—22no—
91#949—21no—
92#942—21no—
93#932—21no—
94#930—21no—
95#929—21no—
96C4×#67—41no—
97#926C2311yescharacteristic
98#924—21no—
99C2×C4×Dic3—21no—
100C2×C4×Dic3—21no—
101C2×C4×Dic3—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×#208C2311yescharacteristic
107C2×#208C2311yescharacteristic
108C2×#208C2311yescharacteristic
109C2×#208C2311yescharacteristic
110#907—41no—
111D3×M16—41no—
112#901—22no—
113C3×#101—21no—
114C3×#100—21no—
115C3×C22⋊C8—22no—
116C3×C22⋊C8—41no—
117#837—21no—
118#837—21no—
119#837—21no—
120#837—21no—
121#837—41no—
122#830—22no—
123#830—41no—
124#825—21no—
125#825—21no—
126#825—21no—
127#825—21no—
128#824—21no—
129#823—21no—
130#822—22no—
131#822—41no—
132#819—41no—
133#815—21no—
134#815D412yescomplemented
135Dic3⋊C8—22no—
136Dic3⋊C8—22no—
137C8×Dic3—21no—
138C8×Dic3D412yescomplemented
139#805—41no—
140#575—31no—
141#568—31no—
142#568C2×D311yescharacteristic
143#525—31no—
144#525C2×D311yescharacteristic
145C22×C42—31no—
146#444—62no—
147#443—62no—
148#432—62no—
149#431—62no—
150#430—31no—
151#430—31no—
152#430—31no—
153#423—31no—
154#423—31no—
155#421—31no—
156#421—31no—
157C2×#104—31no—
158C2×#104—31no—
159C2×#104—31no—
160C2×#104—31no—
161#413—31no—
162#413C2×D311yescharacteristic
163#407—31no—
164#407—31no—
165#407C2×D311yescharacteristic
166#407C2×D311yescharacteristic
167C4×M16—31no—
168C4×M16—31no—
169#344—31no—
170#344—31no—
171#344C2×D311yescharacteristic
172#344C2×D311yescharacteristic
173#339—31no—
174#339—31no—
175#339—31no—
176#339—31no—
177#338—31no—
178H(3,ℤ/4)—31no—
179C24×C3#4111yescharacteristic
180C3×#41—21no—
181C3×#41—21no—
182C3×#41—21no—
183C3×#41—21no—
184C2×C3×Q8#4111yescharacteristic
185C2×C3×D4—21no—
186C2×C3×D4—41no—
187C2×C3×D4#4111yescharacteristic
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4#4111yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C2×#67—21no—
198C2×#67—41no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—44no—
206#235—21no—
207#235—21no—
208#235—21no—
209#235—21no—
210C2×D12#4111yescharacteristic
211C2×C4×D3—21no—
212C2×Dic6#4111yescharacteristic
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—21no—
216C3×M16—21no—
217C3×M16—21no—
218C3×M16—21no—
219C3×M16—21no—
220C3×M16—21no—
221C3×M16—42no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—42no—
226C2×C3×C8C2×D412yes—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—41no—
229C3×C22⋊C4—21no—
230C3×C22⋊C4—41no—
231C3×C22⋊C4—41no—
232C3×C42—21no—
233#217—21no—
234#217—41no—
235#216—22no—
236#215—22no—
237#214—22no—
238#213—22no—
239(C2×D3)⋊C4—41no—
240C4⋊Dic3—21no—
241Dic3⋊C4—41no—
242C4×Dic3—21no—
243C4×Dic3—21no—
244C4×Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3C2×D411yescharacteristic
250C4×Dic3C2×D411yescharacteristic
251C4×Dic3#4111yescharacteristic
252C4×Dic3#4111yescharacteristic
253#208—21no—
254#208—21no—
255#208—21no—
256#208—21no—
257#208—21no—
258#208—21no—
259#208—21no—
260#208—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—21no—
264C2×C3⋊C8C2×D412yes—
265ΓL(1,𝔽25)—42no—
266D3×C8—42no—
267C2×#41—31no—
268C2×#41C22×D311yescharacteristic
269C23×C4—61no—
270C23×C4C22×D311yescharacteristic
271#137—61no—
272#137—61no—
273AGL(1,ℤ/8)—61no—
274AGL(1,ℤ/8)—61no—
275#135—62no—
276#131—22no—
277#131—41no—
278#131—62no—
279#131—121no—
280C2×M16—31no—
281C2×M16—31no—
282C2×M16—31no—
283C2×M16—31no—
284C2×M16—41no—
285C2×M16—121no—
286C2×M16C22×D311yescharacteristic
287C2×M16C22×D311yescharacteristic
288C2×M16C22×D311yescharacteristic
289C2×M16C22×D311yescharacteristic
290#123—21no—
291#123—61no—
292C22⋊Q8—21no—
293C22⋊Q8—61no—
294#121—21no—
295#121—61no—
296#120—21no—
297#120—61no—
298C4×D4—41no—
299C4×D4—121no—
300#117—31no—
301#117C22×D311yescharacteristic
302C2×C42—31no—
303C2×C42—31no—
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42—61no—
308C2×C42—61no—
309#108—62no—
310#108—121no—
311#108—121no—
312C4⋊C8—62no—
313C4⋊C8—62no—
314#104—61no—
315#104—61no—
316#104—61no—
317#104—61no—
318#104—61no—
319#104—61no—
320#104—61no—
321#104—61no—
322#104—121no—
323#104—121no—
324#101—21no—
325#101—61no—
326#100—21no—
327#100—61no—
328C22⋊C8—22no—
329C22⋊C8—41no—
330C22⋊C8—62no—
331C22⋊C8—121no—
332#97—32no—
333#97—61no—
334C4×C8—32no—
335C4×C8—61no—
336C23×C3—21no—
337C23×C3—21no—
338C23×C3—21no—
339C23×C3—21no—
340C23×C3—42no—
341C23×C3C2×#4111yescharacteristic
342C22×D3—21no—
343C3×Q8—21no—
344C3×Q8—21no—
345C3×D4—21no—
346C3×D4—21no—
347C3×D4—41no—
348C3×D4—41no—
349C3×D4—82no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C2×#4111yescharacteristic
363C2×C3×C4C2×#4111yescharacteristic
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4H(5,𝔽2)11yescharacteristic
366C2×C3×C4H(5,𝔽2)11yescharacteristic
367#67—41no—
368#67—41no—
369#67—82no—
370C2×Dic3—24no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—21no—
376C2×Dic3—42no—
377C2×Dic3—42no—
378C2×Dic3—42no—
379C2×Dic3—42no—
380C2×Dic3—42no—
381C2×Dic3—41no—
382D12—21no—
383D12—21no—
384C4×D3—41no—
385C4×D3—41no—
386Dic6—21no—
387Dic6—21no—
388C3×C8—22no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—22no—
393C3⋊C8—41no—
394C3⋊C8—41no—
395C3⋊C8—41no—
396C24#23511yescharacteristic
397#41—21no—
398#41—21no—
399#41—21no—
400#41—21no—
401#41—61no—
402#41—61no—
403#41—61no—
404#41—61no—
405C2×Q8—31no—
406C2×Q8#23711yescharacteristic
407C2×D4—21no—
408C2×D4—31no—
409C2×D4—41no—
410C2×D4—61no—
411C2×D4—121no—
412C2×D4#23711yescharacteristic
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—124no—
428C22×C4#23511yescharacteristic
429C22×C4C23×D311yescharacteristic
430Q16—62no—
431SD16—62no—
432SD16—62no—
433D8—62no—
434M16—21no—
435M16—21no—
436M16—21no—
437M16—21no—
438M16—21no—
439M16—21no—
440M16—21no—
441M16—21no—
442M16—42no—
443M16—61no—
444M16—61no—
445M16—61no—
446M16—61no—
447M16—61no—
448M16—61no—
449M16—61no—
450M16—61no—
451M16—122no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—32no—
456C2×C8—42no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—122no—
461C2×C8D3×D412yes—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—61no—
465C4⋊C4—121no—
466C22⋊C4—21no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—61no—
470C22⋊C4—121no—
471C22⋊C4—121no—
472C42—21no—
473C42—31no—
474C42—31no—
475C42—31no—
476C42—31no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—61no—
482C42—61no—
483C42—61no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3—41no—
492C22×C3—41no—
493C22×C3—81no—
494C22×C3#52511yescharacteristic
495C22×C3#55111yescharacteristic
496C22×C3#55111yescharacteristic
497C2×D3—41no—
498C2×D3—81no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—81no—
502C3×C4#52511yescharacteristic
503C3×C4#53511yescharacteristic
504C3×C4#55311yescharacteristic
505C3×C4#55311yescharacteristic
506Dic3—22no—
507Dic3—22no—
508Dic3—41no—
509Dic3—41no—
510Dic3—41no—
511Dic3—81no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—42no—
517C23—61no—
518C23C2×#23511yescharacteristic
519Q8—21no—
520Q8—21no—
521Q8—61no—
522Q8—61no—
523D4—21no—
524D4—21no—
525D4—41no—
526D4—41no—
527D4—61no—
528D4—61no—
529D4—82no—
530D4—121no—
531D4—121no—
532D4—242no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—64no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—122no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4—122no—
554C2×C4—122no—
555C2×C4—121no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C2×#23711yescharacteristic
560C2×C4#100411yescharacteristic
561C2×C4#100411yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C8—22no—
565C8—41no—
566C8—41no—
567C8—41no—
568C8—62no—
569C8—121no—
570C8—121no—
571C8—121no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—42no—
576C2×C3—81no—
577C2×C3#259811yescharacteristic
578D3—81no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—81no—
589C22—121no—
590C22—241no—
591C22#549511yescharacteristic
592C22#554711yescharacteristic
593C22#555611yescharacteristic
594C4—41no—
595C4—41no—
596C4—62no—
597C4—62no—
598C4—81no—
599C4—121no—
600C4—121no—
601C4—121no—
602C4—241no—
603C4#559211yescharacteristic
604C4#560111yescharacteristic
605C4#560611yescharacteristic
606C4#566111yescharacteristic
607C3#2938411yescharacteristic, complemented, Sylow
608C2—21no—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—81no—
613C2—241no—
614C2#7477611yescharacteristic
subgroupG/H
centerC4#5592
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1891C2
socleC2×C3#2598
prime radical#71715C2
hypercenter#1891D3
hypocenterC3#29384
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1891

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #83811C2semidirect
2. #78619C2semidirect
3. C3×#1891C2semidirect
4. #72028C2semidirect
6. #71711C2semidirect
7. #70144C2semidirect
8. #70130C2semidirect
9. #69992C2semidirect
10. #69987C2semidirect
11. #69965C2semidirect
12. #69033C2semidirect
13. #69024C2semidirect
14. #69017C2semidirect
15. #69012C2semidirect
67. #1891D3semidirect
134. #815D4semidirect
138. C8×Dic3D4semidirect
607. C3#29384semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#838112
#786192
C3×#18912
#720282
#717152
#717112
#701442
#701302
#699922
#699872
#699652
#690332
#690242
#690172
#690122
#293843